Problem: Factor the following expression: $-9$ $x^2$ $-34$ $x+$ $8$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-9)}{(8)} &=& -72 \\ {a} + {b} &=& & & {-34} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-72$ and add them together. Remember, since $-72$ is negative, one of the factors must be negative. The factors that add up to ${-34}$ will be your ${a}$ and ${b}$ When ${a}$ is ${2}$ and ${b}$ is ${-36}$ $ \begin{eqnarray} {ab} &=& ({2})({-36}) &=& -72 \\ {a} + {b} &=& {2} + {-36} &=& -34 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-9}x^2 +{2}x {-36}x +{8} $ Group the terms so that there is a common factor in each group: $ ({-9}x^2 +{2}x) + ({-36}x +{8}) $ Factor out the common factors: $ x(-9x + 2) + 4(-9x + 2) $ Notice how $(-9x + 2)$ has become a common factor. Factor this out to find the answer. $(-9x + 2)(x + 4)$